Coherent Quantum Information Transfer Between Topological And Conventional Qubits

ABSTRACT

Computing bus devices that enable quantum information to be coherently transferred between topological and conventional qubits are disclosed. A concrete realization of such a topological quantum bus acting between a topological qubit in a Majorana wire network and a conventional semiconductor double quantum dot qubit is described, The disclosed device measures the joint (fermion) parity of the two different qubits by using the Aharonov-Casher effect in conjunction. with an ancillary superconducting flux qubit that facilitates the measurement. Such a parity measurement, together with the ability to apply Hadamard gates to the two qubits, allows for the production of states in which the topological and conventional qubits are maximally entangled, and for teleporting quantum states between the topological and conventional quantum systems.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of provisional U.S. patentapplication Ser. No. 61/412,225, filed Nov. 10, 2010, the disclosure ofwhich is hereby incorporated herein by reference.

This application is related to U.S. patent application [attorney docketnumber MSRD_(—)331530.03], filed on even date herewith, entitled“Coherent Quantum Information Transfer Between Conventional Qubits,” thedisclosure of which is hereby incorporated herein by reference.

TECHNICAL FIELD

Generally, this application relates to quantum computational systems.More specifically, the application relates to devices that can entangleand coherently transfer quantum information between topological andconventional quantum media.

BACKGROUND

The topological approach to quantum information processing obtainsexceptional fault-tolerance by encoding and manipulating information innon-local (i.e., topological) degrees of freedom of topologicallyordered systems. Such non-local degrees of freedom typically do notcouple to local operations. Consequently, the error rates fortopological qubits and computational gates may be exponentiallysuppressed with distance between anyons and inverse temperature.

This suppression of error rates may provide an advantage overconventional quantum computing platforms. However, it also makes itchallenging to coherently transfer quantum information into and out oftopological systems. Not only is coupling the non-local degrees offreedom in the topological system to an external system required, but itmust be done in a controlled and coherent manner. Consequently, it isdesirable to create quantum entanglement between the topological andconventional states.

SUMMARY

Described herein is a device that can entangle and coherently transferquantum information between topological and conventional quantum media.Such a device may be referred to herein as a “topological quantum bus.”The device allows for harnessing the relative strengths of the differentquantum media, and may be implemented in connection with quantumcomputation.

An example of how a topological quantum bus may be useful stems from theunderstanding that a computationally universal gate set cannot beproduced for Ising anyons using topologically-protected braidingoperations alone. Unless a truly topologically ordered Ising system isprovided (which is not the case for superconductor-based systems,including Majorana wires), and can perform certain well-known topologychanging operations, braiding operations need to be supplemented withtopologically unprotected operations. Fortunately, these can beerror-corrected for an error-rate threshold of approximately 0.14 byusing the topologically-protected Ising braiding gates to perform“magic-state distillation.”

Within a topological system, unprotected gates can be generated by, forexample, bringing non-Abelian anyons close to each other, whichgenerically splits the energy degeneracy of the topological state spaceand, hence, dynamically gives rise to relative phases, or by usinginterfering currents of anyons, which can have an equivalent effect.However, besting even such a high error threshold may still provedifficult using unprotected operations within a topological system, as aresult of significant non-universal effects. A topological quantum busallows for the desired topologically unprotected gates to be importedfrom conventional quantum systems, for which error rates below 0.14% oflogical operations have already been achieved.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 depicts a semiconductor nanowire coupled by proximity with ans-wave superconductor in the presence of an in-plane magnetic field.

FIG. 2 is a plot of energy dispersion for the semiconductor depicted inFIG. 1, with spin-orbit coupling in the magnetic field.

FIG. 3 is a schematic of a device for entangling topological andconventional qubits.

FIG. 4 is a schematic of a device for coherent quantum informationtransfer between topological and conventional semiconductor qubits usingjoint parity measurements.

FIG. 5 is a flowchart of a method for coherent quantum informationtransfer between topological and conventional semiconductor qubits usingjoint parity measurements.

DETAILED DESCRIPTION

A method for implementing a topological quantum bus may include the useof measurements in an entangled basis, e.g., Bell state measurements.For a topological quantum bus, this can be achieved by a measurement ofthe joint parity of a topological-conventional qubit pair, given theability to perform Hadamard gates on any qubit.

Joint parity measurement corresponds to the two orthogonal projectors:

Π₀=|00

00|+|11

11|,  (1)

Π₁=|01

01|+|10

10|,  (2)

where |0

and |1

and are the logical basis states of the qubits. Topological systems,however, tend to be rather obstructive to such hybridization withexternal systems. For example, quantum Hall states (the archetypaltopological systems) require a large background magnetic field, which“destroys” superconductivity and eliminates the possibility of couplingto Josephson-junction qubits.

Fortunately, recently proposed implementations of Majorana nanowiresappear promising for overcoming such obstacles. These wires localizezero energy Majorana fermions at their endpoints and, as such, provide aone-dimensional topologically protected two-level system. It may appearthat such a system might provide a topological qubit, but lack quantuminformation processing. However, a network of Majorana wires may beformed, and manipulated using gate electrodes in a manner that performsbraiding exchanges of their endpoints (and, hence, their respectiveMajorana fermions). This generates the topologically-protected braidingoperations of Ising anyons (up to an overall phase) on the topologicalstate space.

It follows from this that Majorana wire networks can be utilized asIsing anyons for topologically-protected quantum information processing.A concrete realization of a topological quantum bus that uses theAharonov-Casher effect to coherently transfer quantum informationbetween a topological qubit in a Majorana wire system and a conventionalsemiconductor double-dot qubit will now be described in detail.

The Aharonov-Casher effect involves interference of particles withmagnetic moment (vortices) moving around a line charge. It may enableperformance of non-local measurements of charge in a region by utilizingthe interference of vortices traveling through two different pathsaround the region. For superconducting systems it is natural to try touse Abrikosov vortices in this context. However, Abrikosov vortices ins-wave superconductors may have rather large mass due to the largenumber of subgap states localized in their cores. As a result, thesevortices may behave classically.

An alternative is to use Josephson vortices (fluxons), which arise dueto phase-slip events in Josephson junctions. Their effective mass isdetermined by the charging and Josephson energies of the junction, andcan be much smaller than that of Abrikosov vortices, allowing them tobehave quantum-mechanically. Indeed, the Aharonov-Casher effect withJosephson vortices has been experimentally observed, and severalproposals have been made to utilize it in the context of topologicalquantum information processing.

The basic element in the implementation of Majorana wires is asemiconductor nanowire with strong spin-orbit inter-actions, coupledwith an s-wave superconductor. FIG. 1 depicts a semiconductor nanowirecoupled by proximity with an s-wave superconductor, in the presence ofan in-plane magnetic field. The Hamiltonian (with =1) for such ananowire is:

0 = ∫ - L / 2 L / 2   x   ψ σ †  ( x )  ( - ∂ x 2 2  m * - μ + α  σ y  ∂ x  + V x  σ x ) σσ ′  ψ σ ′  ( x ) , ( 3 )

where m*, μ, and α are the effective mass, chemical potential, andstrength of spin-orbit Rashba interaction, respectively, and L is thelength of the wire, which is taken to be much longer than the effectivesuperconducting coherence length ξ in the semiconductor.

An in-plane magnetic field B_(x) leads to spin splittingV_(x)=g_(SMμB)B_(x)/2, where g_(SM) is the g-factor in thesemiconductor, and μ_(B) is the Bohr magneton. FIG. 2 is a plot ofenergy dispersion for the semiconductor, with spin-orbit coupling in themagnetic field B_(x), which opens a gap in the spectrum. When thechemical potential μ is in this gap, the nanowire coupled with thes-wave superconductor is driven into topological phase.

In other words, when coupled with an s-wave superconductor, the nanowirecan be driven into a non-trivial topological phase with Majoranazero-energy states localized at the ends when the chemical potential isproperly adjusted and lies in the gap. In the simplest case of asingle-channel nanowire, the topological phase corresponds to|V_(x)|>√{square root over (μ²+Δ²)}, where Δ is the proximity-inducedpairing potential. Multi-channel use is described in R. M. Lutchyn, T.Stanescu, and S. Das Sarma, Phys. Rev. Lett. 106, 127001 (2011),arXiv:1008.0629, incorporated herein by reference.

As seen in FIGS. 3 and 4, two Majorana fermions γ₁ and γ₂ residing atthe ends of a wire 10 constitute a topological qubit 20, since they giverise to a two-level system that is degenerate up to O (e^(−L/ξ))corrections that are exponentially suppressed with the length of thewire 10. Indeed, a non-local Dirac fermion operator can be formallydefined as c=′γ₁+i′γ₂, and then the two logical states of the qubitcorrespond to the state in which this Dirac fermion is unoccupied |0

≡|n_(p)=0

and occupied |1

≡|n_(p)=1

, where c|n_(p)=1

=n_(p)=0

, c|n_(p)=0

=0, and c^(†)c|n_(p)=n_(p)|n_(p)

. Thus, the topological qubit states are characterized by fermion parityn_(p)=0, 1. As previously mentioned, in a network of such wires, theseMajorana fermions γ₁ and γ₂ behave as Ising non-Abelian anyons when theyare translocated, e.g., using gate electrodes.

Topological and conventional qubits can be entangled by measuring thefermion parity on the superconducting island using the Aharonov-Cashereffect. FIG. 3 is a schematic of a device for entangling a topological(e.g., Majorana wire) qubit 20 and a conventional (e.g., semiconductordouble-dot) qubit 30. A flux qubit Φ having three Josephson junctions(the strips labeled J₁, J₂, and J₃) supports clockwise orcounter-clockwise supercurrent. When E_(J1)=E_(J3), there isinterference between quantum phase slips across junctions J₁ and J₃.These phase slips correspond to Josephson vortex tunneling encirclingthe superconducting islands as shown by the dashed line. Via theAharonov-Casher effect, quantum interference of vortices around theislands produces an energy splitting for the flux qubit (at itsdegeneracy point) that strongly depends on the state of the topologicaland conventional semiconductor qubits 20, 30. The nanowire 10 may havetopological 10T and non-topological 10N segments. The latter can beachieved by driving the wire 10 into the insulating or trivialsuperconducting phases.

More specifically, consider the superconducting flux qubit Φ withJosephson junctions designed to have left-right symmetry such thatJosephson coupling energies E_(J1)=E_(J3)≡E_(J). The twocurrent-carrying states, clockwise |

and counter-clockwise |

, form the basis states of the flux qubit Φ. When the applied externalflux piercing the flux qubit Φ is equal to a half flux quantum, i.e.,Φ=h/4 e, there is a degeneracy between the two current-carrying states.This degeneracy is lifted by the macroscopic quantum tunneling betweenthe state |

and |

due to the presence of a finite charging energy of the islands, whichtends to delocalize the phase. Thus, the new eigenstates of the qubitare |±

=(|

±|

)/√{square root over (2)}. For the device shown in FIG. 3, the energysplitting between states |±

depends on the quantum interference of the fluxon trajectories. Indeed,the total Josephson energy of the qubit is

$\begin{matrix}{{\frac{U_{J}}{E_{J}} = {- \left\lbrack {{\cos \; \phi_{1}} + {\cos \; \phi_{2}} + {\frac{E_{J_{2}}}{E_{J}}{\cos \left( {{2\; \pi \; \frac{\Phi}{\Phi_{0}}} - \phi_{1} - \phi_{2}} \right)}}} \right\rbrack}},} & (4)\end{matrix}$

where it is assumed that E_(J1)>E_(J), in contrast with values typicallyused for flux qubits.

The potential U_(J) reaches its minima at two inequivalent points(φ₁,φ₂)=(±φ*+2πm,∓φ*∓2πn) for a given n and m which correspond toclockwise and counter-clockwise circulating currents, and(φ*=cos⁻¹(E_(J)/E_(J2)). Starting, for example, from the configurationwith (φ*−φ*), there are two paths to tunnel to a different flux state:(φ*,−φ*)→(φ*−2π,φ*), and (φ*,−φ*)→(−φ*,φ*+2π), which correspond to aphase slip through junction J₁ or J₃, respectively. As a result, thereis an interference between the two paths that encircle the middleislands in the system shown in FIG. 3. Note that the amplitude for thephase slips across the middle junction is suppressed in this setup sinceE_(J2)>E_(J). This interference is sensitive to the total chargeenclosed by the paths, i.e., the charge residing on the two islands, andis determined by the Berry phase contribution.

For the device shown in FIG. 3, the splitting energy is given byΔ=Δ₀cos(Ø_(AC)/2), where Ø_(AC)=πq/e is the Aharonov-Casher phase fortotal charge on the islands given by q=en_(p)+q_(ext), where n_(p) isthe fermion occupation of the Majorana wire 10 and q_(ext) is theinduced gate charge on the islands. Given that the qubit splittingenergy now depends on the fermion occupation number, the state of atopological qubit can be efficiently read out using, for example, thewell-known radio-frequency (RF) reflectometry technique, which can becarried out with sub-microsecond resolution times. It is assumed thatsuperconducting islands have the same charging energy yielding the sametunneling amplitude Δ₀. Assuming E_(J)/E_(C)≈10 and E_(J2)/E_(J)≈1.25,WKB approximation gives Δ₀≈0.02 hv_(a), where v_(a) is the attemptfrequency, which is estimated to be v_(a)˜0.1−1 GHz.

A situation where q_(ext) has a quantum component corresponding tocoherent electron tunneling inside the area enclosed by the vortexcirculation can be realized, for example, by coupling the flux qubit Φto a semiconductor double quantum dot (DQD) qubit 30, as shown in FIG.3. Galvanic isolation may exist between the superconductor andsemiconductor so that there is no charge transfer between them. DQDqubits may be realized using indium arsenide (InAs) nanowires, which maythus serve as a dual-purpose component (i.e., also being used for theMajorana nanowires).

If there is a single electron in the DQD, the logical qubit basis statescan be defined to be |0

≡|0

_(U)

|1

_(L), where the electron occupies the lower quantum dot, and |1

≡|01

_(U)

|0

_(L), where the upper quantum dot is occupied. This situationcorresponds to a semiconductor charge qubit. If there are two electronsin the DQD, then the logical qubit basis states can be defined to be |0

≡|0

_(U)

|2

_(L) and |1

≡|1

_(U)

|1

_(L), where the electron spins are in the singlet and triplet states,respectively. This situation corresponds to the semiconductor spinqubit.

Both these qubits share a common feature that can be exploited: thequbit basis states correspond to the electron parity on the upper dotenclosed by the vortex circulation. If the evolution of thesemi-conductor qubit is much slower than the measurement time and fluxontunneling rate, then the flux qubit Φ can be used to entangletopological and conventional qubits 20, 30 via the Aharonov-Cashereffect. Indeed, the flux qubit splitting energy Δ is the same forcombined topological-DQD qubit states with equal joint-parity, i.e., thecombined states |00

and |11

have the same splitting, and |01

and |10

have the same splitting. Thus, measurement of the flux qubit splittingenergy Δ is equivalent to a joint parity measurement corresponding tothe projectors Π₀ and Π₁ from Eqs. (1) and (2) acting on thetopological-DQD qubit pair.

If the topological and conventional qubits 20, 30 are initially preparedin the superposition states |ψ_(T)

=α_(T)|0

+β_(T)|1

and |ψ_(C)

=α_(C)|0

+β_(C)|1

, respectively, then application of the even or odd parity projectorsgives the (unnormalized) states

Π₀([ψ_(T))

|ψ_(C)

)=α_(T)α_(C)|00

+β_(T)β_(C)11

  (5)

Π₁([ψT)

|ψC

)=α_(T)β_(C)|01

+β_(T)α_(C)10

,  (6)

It should be understood that the flux qubit Φ acts as an interferometerthat enables this measurement.

Qubits can be entangled and coherent quantum information transferperformed using parity measurements with the help of two flux qubits.The maximally entangled Bell states (which can be used as entanglementresources) may be denoted as

|Φ_(μ)

≡(

σ_(μ))(|01

−|10

)√{square root over (2)},  (7)

for μ=0,1,2,3(σ_(μ)=

). The ability to perform measurements in the Bell basis allows for theteleportation of quantum states, and hence, for the transfer quantuminformation. It should be understood from Eqs. (5) and (6) that jointparity measurements can produce entangled states, such as Bell states.More generally, however, it should be understood that

Π₀=|Φ

Φ₁|+|Φ₂

Φ₂|  (8)

Π₁=|Φ

Φ₀|+|Φ₃

Φ₃|  (9)

Π^(˜) ₀=(H

H)Π₀(H

H)=|Φ₂

Φ₂|⇄|Φ₃

Φ₃|  (10)

Π^(˜) ₁=(H

H)Π₁(H

H)=|Φ₀

Φ₂|⇄|Φ₃

Φ₃|  (11)

where the (single-qubit) Hadamard gate is given by

$\begin{matrix}{H = {{\frac{1}{\sqrt{2}}\begin{bmatrix}1 & 1 \\1 & {- 1}\end{bmatrix}}.}} & (12)\end{matrix}$

Hence, joint parity measurements combined with Hadamard gates fullyresolves the Bell basis.

Hadamard gates can be generated (with topological protection) bybraiding Ising anyons and through standard methods for conventionalqubits. As described above, the device depicted in FIG. 3 can be used toimplement a joint parity measurement of a topological-conventional qubitpair 20, 30. It can also be used to implement joint parity measurementsof topological-topological and conventional-conventional qubit pairs.

Specifically, consider the device 100 shown in FIG. 4 where there arethree flux qubits Φ₁, Φ₂, and Φ₃. One of these, Φ₃, is coupled to twosemi-conductor DQD qubits 30A, 30B. Again, it is assumed that there isright-left symmetry (E_(J7)=E_(J9)) so that fluxon tunneling in thesuperconducting qubit that allows for measuring the combined chargeparity for the conventional-conventional qubit pair, as explained above.The other flux qubit Φ₂, with (E_(J4)=E_(J6)) allows for joint paritymeasurements on topological-topological qubit pairs. The combined device100 allows quantum information to be transferred between topological andconventional qubits 20, 30. Finally, by tuning the external fluxes Φaway from the degeneracy point, flux and conventional or topologicalqubits can be decoupled.

It should be understood that the joint parity measurement device 100depicted in FIG. 4 not only allows for coherent entanglement andtransfer of information between topological and conventional systems,but also provides a new method of entangling conventional qubits, e.g.,semiconductor charge or spin qubits, with each other, and hence couldalso be useful for purely conventional systems.

It should be understood from the foregoing description that atopological quantum bus may process is accordance with the method 200 ofFIG. 5. That is, a method 200 for coherent coupling of topological andconventional qubits may include, at step 202, providing a topologicalqubit and a conventional qubit that are coupled to one another via aplurality of Josephson junctions.

At 204, a joint parity measurement maybe performed. The joint paritymeasurement may enable quantum information to be coherently transferredbetween the topological and conventional qubits. The joint parity of thetopological and conventional cubits may be measured via theAharonov-Casher effect. The joint parity measurement may be performed inthe Bell basis. Hadamard gates may be applied to the topological andconventional qubits. The Hadamard gates may be generated by braidingIsing anyons.

At 206, quantum information may be coherently transferred between thetopological and conventional qubits.

1. A system for coherent coupling of topological and conventional qubits, the system comprising: first, second, and third Josephson junctions; a superconducting flux qubit that supports clockwise and counterclockwise supercurrents in a superconducting ring defined by the first, second, and third Josephson junctions, the flux qubit having a first basis state that corresponds to the clockwise supercurrent and a second basis state that corresponds to the counterclockwise supercurrent; a Majorana nanowire network coupled to a first superconducting island formed between the first and second Josephson junctions, wherein the Majorana nanowire network provides a topological qubit; a semiconductor coupled to a second superconducting island formed between the second and third Josephson junctions, wherein the semiconductor provides a conventional double-dot qubit, wherein a joint parity of the topological and conventional qubits is measured via an Aharonov-Casher effect that gives rise to an energy splitting in the flux qubit that depends on a total charge enclosed in a region between the first and third Josephson junctions.
 2. The system of claim 2, wherein the energy splitting in the flux qubit yields a measurement of fermion parity enclosed in the region.
 3. The system of claim 2, wherein the fermion parity measurement provides the joint parity measurement of the topological and conventional qubits.
 4. The system of claim wherein, when a Josephson coupling energy at the first Josephson junction equals a Josephson junction energy at the. third Josephson junction, interference occurs between the first and third Josephson junctions.
 5. The system of claim 1, wherein a quantum state is produced such that the topological and conventional qubits are entangled.
 6. A topological quantum bus, comprising: a first device that performs a joint parity measurement on a topological-conventional qubit pair; a second device that performs a joint parity measurement on a topological-topological qubit pair; and a third device that performs a joint parity measurement on a conventional-conventional qubit pair, wherein a first of the topological cubits and a first of the conventional qubits are coherently coupled, and quantum information is transferred between the first topological qubit and the first conventional qubit.
 7. The topological quantum bus of claim 6, further comprising; means for performing a joint parity measurement via an Aharonov-Casher effect that enables the quantum information to be coherently transferred between the first topological qubit and the first conventional qubit.
 8. The topological quantum bus of claim 6, wherein a quantum state is produced such that the first topological qubit and the first conventional qubit are entangled.
 9. The topological quantum bus of claim 6, wherein a quantum state is teleported between the topological and conventional qubits.
 10. The topological quantum bus of claim 6, wherein the first device comprises a first superconducting flux qubit that supports clockwise and counterclockwise supercurrents in a first superconducting ring surrounding a region containing the first topological qubit and the first conventional qubit, wherein the second device comprises a second superconducting flux qubit that supports clockwise and counterclockwise supercurrents in a second superconducting ring surrounding a region containing second and third of the topological qubits, and wherein the third device comprises a third superconducting flux qubit that supports clockwise and counterclockwise supercurrents in a third superconducting ring surrounding a region containing second and third of the conventional qubits.
 11. The topological quantum bus of claim 10, wherein the first device comprises first, second, and third Josephson junctions that define the first superconducting ring, a first Majorana wire network coupled to a first superconducting island disposed between the first and second Josephson junctions, and a first semiconductor coupled to a second superconducting island disposed between the second and third Josephson junctions, wherein the second device comprises fourth, fifth and sixth Josephson junctions that define the second superconducting ring, a second Majorana wire network coupled to a third superconducting island disposed between the fourth and fifth Josephson junctions, and a third Majorana wire network coupled to a fourth superconducting island disposed between the fifth and sixth Josephson junctions, and wherein the third device comprises seventh, eighth, and ninth Josephson junctions that define the third superconducting ring, a second semiconductor coupled to a fifth superconducting island disposed between the seventh and eighth Josephson junctions, and a third semiconductor coupled to a sixth superconducting island disposed between the eighth and ninth Josephson junctions.
 12. A method for coherent coupling of topological and conventional qubits, the method comprising: providing a topological qubit, providing a conventional qubit; and coherently coupling and transferring quantum information between the topological and conventional qubits.
 13. The method of claim 12, further comprising: performing a joint parity measurement that enables the quantum information to be transferred between the topological and conventional qubits.
 14. The method of claim 13, wherein the joint parity of the topological and conventional qubits is measured via a Aharonov-Casher effect.
 15. The method of claim 14, wherein the Aharonov-Casher effect gives rise to an energy splitting in a flux qubit that depends on a total charge enclosed in a region proximate a flux qubit.
 16. The method of claim 15, wherein the energy splitting in the flux qubit yields a measurement of fermion parity enclosed in the region.
 17. The method of claim 16, wherein the fermion parity measurement provides the joint parity measurement of the topological and conventional qubits.
 18. The method of claim 12, further comprising: teleporting a quantum state between the topological and conventional qubits.
 19. The method of claim 12, further comprising: producing a quantum state such that the topological and conventional qubits are entangled.
 20. The method of claim 12, further comprising: producing a quantum state such that the topological and conventional qubits are maximally entangled. 